Every group admits an initial homomorphism to a pi-powered group

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Statement

Suppose G is a group and \pi is a set of primes. There exists a \pi-powered group K and a homomorphism of groups \alpha:G \to K such that for any \pi-powered group L and any homomorphism \varphi:G \to L, there is a unique homomorphism \theta:K \to L such that \varphi = \theta \circ \alpha.

We can think of the functor that sends G to the group K as the free \pi-powering functor. It is the left-adjoint functor to the forgetful functor from \pi-powered groups to groups.

In the localization terminology

The term \pi-local is sometimes used to refer to a group that is powered over all the primes not in \pi. The functor described above is, in that context, termed the \pi'-localization functor. By \pi' we mean the complement of \pi in the set of all primes.

Related facts

Embeddability results

We are interested in cases where the canonical homomorphism from the group to its \pi-powering is injective, and in related questions, some of which are explored below.

Group assumption on both the starting group and the big group Divisibility/powering/torsion assumption on the starting group Divisibility/powering/torsion assumption on the big group Is this always possible? Proof
nilpotent group none divisible group No nilpotent group need not be embeddable in a divisible nilpotent group
nilpotent group \pi-torsion-free group (equivalently, \pi-powering-injective group) \pi-powered group Yes every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group
arbitrary group none divisible group Yes every group is a subgroup of a divisible group, every pi-group is a subgroup of a divisible pi-group
arbitrary group \pi-torsion-free group (equivalently, \pi-powering-injective group) \pi-powered group No Powering-injective group need not be embeddable in a rationally powered group